Complete characterization of convergence to equilibrium for an inelastic Kac model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of aMaxwellian gas to inelastic particles.We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index α is determined by the so-called degree of inelasticity,p >0, of the particles: α = 2 1+p . This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index α. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.